England's defeat at the hands of Ireland earlier in the day meant that any win would seal a runners-up spot for Warren Gatland's side, and they ground out an ugly triumph despite being second best for much of the contest.
France could - and should - have taken the lead in the second half when they were awarded a penalty right in front of the post, but Francois Trinh-Duc missed the kick which would have made it 14-16.
It proved to be a costly miss too as Wales hung on, having built a 14-10 lead at half time largely thanks to Liam Williams's opening try.
Gael Fickou responded with a try-of-the-tournament contender at the other end, but ultimately it was France's inaccuracy from the tee which saw them slump to defeat in Cardiff.
The result sees France finish fourth in the final standings, while Wales only miss out to Grand Slam champions Ireland.